Hidden Assemption

[lama]

One of the most devastating things in the Language of Mathematics and its logical reasoning is a hidden assumption, and the worst thing is a first-order hidden assumption.

 

Let FOHA be a First-Order Hidden Assumption.

 

Let UQ be Universal-Quantification.

 

Is there a FOHA in the way we use UQ concept?

 

The UQ is based on the term ‘For All’.

 

The meaning of the word ‘All’ is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection, where in the level of SUM we are no longer in the level of each single element that exists in this collection.

 

The SUM of a collection is actually its Cardinality.

 

If we define some property that can be found in each single element in this collection, then this property cannot be SUMMERIZED, because this definition can exist only in the level of each single element.

 

For example: All men are mortal.

 

The property of being mortal does not changed by the quantity of men.

 

 

In this case we cannot use the Quantity concept because the existence of this definition cannot be developed beyond the Quantity 1 (each single element).

 

So we see that there can be found some common property to several distinguished elements, but this common property is not related to the Quantity concept, and therefore it cannot be SUMMERIZED, or in other words, this common property cannot be summarized beyond the quantity 1, and the meaning of quantity 1 in this case is synonym to the term ‘There exists’.

 

So we can clearly see that a common property that cannot be summarized is actually based on the term ‘There exists’, where this common property’s existence is in the level of each distinguished element that exists in the collection.

 

Let FA be ‘For All’.

 

Let TE be ‘There Exists’.

 

It is very important not to mix between FA and TE.

 

Now let us say that we have a collection of infinitely many elements.

 

For example, the collection of the Natural numbers.

 

The existence of this collection is based on these definitions:

 

a) 1 is a Natural number

 

b) N is a container of only Natural numbers

 

c) 1 is in N

 

d) n = 1

 

e) If n is in N then n+1 is in N.

 

By term (e) we know that for each arbitrary single element in N there is another element that is bigger by 1 from this arbitrary single element.

 

In this case the common concept (that is actually based on TE) is no other then the “famous” Successor concept.

 

Since the Successor is a TE product, we cannot conclude anything about the complete Quantity of N collection.

 

Furthermore, we cannot use this definition:

 

For All n in N, If n is in N then n+1 is in N.

 

We can use FA only if we can summarize the elements of N, but since there are infinitely many elements, then their SUM cannot be found and this if a fundamental difference between a collection of finitely many elements (that its SUM can be found) and a collection of infinitely many elements (that its SUM cannot be found).

 

Since the meaning of the word ‘All’ (in the ‘For All’ term) is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection.

 

But then we can clearly see that by using the prefix ‘For All n in N’ we actually use a hidden assumption that gives us the illusion that we can define the Cardinality of a non-finite collection.

 

Some one can say:

 

But we can use 1-1 mapping between collections of elements, and by this technique we can find if there is or there is no difference in the quantity between the collections, even if we do not know the exact SUM (the Cardinality) of each collection.

 

Let us check this claim:

 

By 1-1 mapping, we take two collections A and B (for example) and try to find a 1-1 map between each element in A and each element in B.

 

If the two collections are finite collections, then we can summarize the Quantity of the 1-1 mapping, and then we compare this result with the cardinality of each examined collection, and only by these 3 Cardinals we can conclude if both collections are Equal to the Cardinal of the 1-1 mapping, or not.

 

If the collections are not equal, then it is obvious that the cardinal (the SUM) of the 1-1 mapping is equal to the smaller collection.

 

But if we use the 1-1 mapping technique between two collections where each one of them has infinitely many elements, then:

 

In this case no cardinality can be found, and our conclusions are depend on TE.

 

But also in this case we want to find out if the two collections have the same quantity of elements, or not.

 

By TE (There Exist) the only information that we can get is only in the level of each single 1-1 mapping, where the SUM of these 1-1 mapping cannot be summarized, because TE cannot go beyond quantity 1.

 

So we clearly can see that if the cardinality of the two examined collections cannot be found, then we cannot use FA (For All) on these collections but only TE (There Exists).

 

Conclusion:

 

We can get a result out of 1-1 mapping technique only if the cardinality of the 1-1 mapping can be found, and it cannot be found if both examined collections are non-finite.

 

 

Now we can clearly see the fundamental conceptual mistake of using FA on a collection of infinitely many elements, which gives us the illusion that we can conclude some meaningful conclusions about the quantitative property of the examined collections.

 

FA (‘For All’) is the Universal Quantifier (which is based on the Quantity concept) and there is a First Order Hidden Assumption in the way we use the Universal Quantifier concept on collections of infinitely many elements.

[JaKiri]

I didn't understand a word of that.

[matt grime]

One of the most devastating things in the Language of Mathematics and its logical reasoning is a hidden assumption' date=' and the worst thing is a first-order hidden assumption.

 

Let FOHA be a First-Order Hidden Assumption.

 

Let UQ be Universal-Quantification.

 

Is there a FOHA in the way we use UQ concept?

 

The UQ is based on the term ‘For All’.

 

The meaning of the word ‘All’ is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection, where in the level of SUM we are no longer in the level of each single element that exists in this collection.[/quote']

 

 

No, that is not true in mathematics as anyone else practices it. In fact as a paragraph in the English lagnauge it is meaningless too.

 

The SUM of a collection is actually its Cardinality.

 

Again, that is not the definition of cardinality.

 

If we define some property that can be found in each single element in this collection, then this property cannot be SUMMERIZED, because this definition can exist only in the level of each single element.

 

again, that makes no sense in any form.

 

For example: All men are mortal.

 

The property of being mortal does not changed by the quantity of men.

 

this isn't a mathematical statement

 

 

In this case we cannot use the Quantity concept because the existence of this definition cannot be developed beyond the Quantity 1 (each single element).

 

So we see that there can be found some common property to several distinguished elements, but this common property is not related to the Quantity concept, and therefore it cannot be SUMMERIZED, or in other words, this common property cannot be summarized beyond the quantity 1, and the meaning of quantity 1 in this case is synonym to the term ‘There exists’.

 

this is a non sequitur.

 

So we can clearly see that a common property that cannot be summarized is actually based on the term ‘There exists’, where this common property’s existence is in the level of each distinguished element that exists in the collection.

 

Let FA be ‘For All’.

 

Let TE be ‘There Exists’.

 

It is very important not to mix between FA and TE.

 

as one is the 'negation' of the other...?

 

Now let us say that we have a collection of infinitely many elements.

 

For example, the collection of the Natural numbers.

 

The existence of this collection is based on these definitions:

 

a) 1 is a Natural number

 

b) N is a container of only Natural numbers

 

c) 1 is in N

 

d) n = 1

 

e) If n is in N then n+1 is in N.

 

By term (e) we know that for each arbitrary single element in N there is another element that is bigger by 1 from this arbitrary single element.

 

In this case the common concept (that is actually based on TE) is no other then the “famous” Successor concept.

 

Since the Successor is a TE product, we cannot conclude anything about the complete Quantity of N collection.

 

who on earth is concluding anything about the "complete quantity"other than that the set is not finite?

 

Furthermore, we cannot use this definition:

 

For All n in N, If n is in N then n+1 is in N.

 

We can use FA only if we can summarize the elements of N, but since there are infinitely many elements, then their SUM cannot be found and this if a fundamental difference between a collection of finitely many element (that its SUM can be found) and a collection of infinitely many elements (that its SUM cannot be found).

 

we are not finding any sums, though. we are not adding anything up.

 

 

Since the meaning of the word ‘All’ (in the ‘For All’ term) is synonym to the word ‘Complete’ and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection, then we can clearly see that by using the prefix ‘For All n in N’ we actually use a hidden assumption that gives us the illusion that we can define the Cardinality of a non-finite collection.

 

again, all and complete are not perfect synonyms in english, and even if they were it isn't the common english meaning that is important. complete has several meanings in mathematics, such as all cauchy sequences have a limit, or all polynomials possess all their roots. and this isn't even a matter of "all" it is a matter of "for all".

 

Some one can say:

 

But we can use 1-1 mapping between collection of elements, and by this technique we can find if there is or there is no difference in the quantity between the collections, even if we do not know the exact SUM (the Cardinality) of each collection.

 

ahem, you aren't acknowledging that the *definition* that two sets have the same cardinality means that there is a bijection between them. the notion of summing the elements of an arbitrary set doesn't even make sense without further explanation.

 

Let us check this clime:

 

"claim", clime means weather or weather conditions

 

By 1-1 mapping, we take two collections A and B (for example) and try to find a 1-1 map between each element in A and each element in B.

 

do you even know the definition of injection?

 

If the two collections are finite collections, then we can summarize the Quantity of the 1-1 mapping, and then we compare this result with the cardinality of each examined collection, and only by these 3 Cardinals we can conclude if both collections are Equal to the Cardinal of the 1-1 mapping, or not.

 

cardinals (infinite ones) belong to strictly higher system than functions since we need to define bijection (or injection by the bernstein-cantor-schroeder theorem) in order to define cardinals for all sets.

 

If the collections are not equal, then it is obvious that the cardinal of the 1-1 mapping is equal to the smaller collection.

 

excuse me? what is the cardinality of a function? i can see how one may choose to define it, but it is always equal to the cardinality of the domain so it is pointless.

 

But if we use the 1-1 mapping technique between two collections where each one of them has infinitely many elements, then:

 

In this no cardinality can be found, and our conclusions are depends on TE.

 

But also in this case we want to find out if the two collections have the same quantity of elements, or not.

 

By TE (There Exist) the only information that we can get is only in the level of each single 1-1 mapping, where the SUM of these 1-1 mapping cannot be summarized, because TE cannot go beyond quantity 1.

 

again this is an undefined use of the word sum.

 

So we clearly can see that if the cardinality of the two examined collections cannot be found, then we cannot use FA (For All) on these collections but only TE (There Exists).

 

Conclusion:

 

We can get a result out of 1-1 mapping technique only if the cardinality of the 1-1 mapping can be found, and it cannot be found if both examined collections are non-finite.

 

more meaningless waffle, I'm afraid.

 

 

Now we can clearly see the fundamental conceptual mistake of using FA on a collection of infinitely many elements, which gives us the illusion that we can conclude some meaningful conclusions about them that is related to the Quantity concept.

 

FA (‘For All’) is the Universal Quantifier (which is based on the Quantity concept) and there is a First Order Hidden Assumption in the way we use the Universal Quantifier concept on collections of infinitely many elements.

 

 

Here's a simple statement:

 

[math] (\forall n \in \mathbb{N})((n>9) \implies (2^n > n^3))[/math]

 

in what way have I done something that is *mathematically* wrong? Not in your opinion that no one else understands, what have I don'e wrong given the correct(ie the ones everyone else uses) meanings of all the terms?

[Dave]

Don't you ever learn, lama? What on earth could possibly posess you to write even more of this stuff when it has been completely ridiculed by everyone on here?

[lama]

]

excuse me? what is the cardinality of a function?

In the case of 1-1 between two collections of Natural numbers' date=' then the cardinality the total number (the SUM) of 1-1 mapping that can be found.

Again, that is not the definition of cardinality.

Yeh, I know The SIZE of a collection is actually its Cardinality.

 

Can you please explain what is the difference between Size of a collection and Sum of a collection?

 

Thank you.

[lama]

Don't you ever learn' date=' lama? What on earth could possibly posess you to write even more of this stuff when it has been completely ridiculed by everyone on here?

[/quote']

Dear Dave,

 

Can you please explain what is the difference between Size of a collection and Sum of a collection?

 

Thank you.

[lama]

Here's a simple statement:

 

[math] (\forall n \in \mathbb{N})((n>9) \implies (2^n > n^3))[/math]

 

in what way have I done something that is *mathematically* wrong? Not in your opinion that no one else understands' date=' what have I don'e wrong given the correct(ie the ones everyone else uses) meanings of all the terms?

[/quote']

The above property is true for each single element > 9.

 

It does not give any information about the number of these elements, thesefore you cannot use the word 'ALL', because it gives you the illusion that you can know the Size of this collection, and I do not see how you can define this Size.

 

So please show us how can we define the Size of N, for example?

 

If you cannot do that, instead "For All", you have to use:

 

"There exists some arbitrary n in N such that ...."

 

Or we can invent a new Quantifeir, which its name is: "For Each" and it is equivalent to ""There exists some arbitrary ...".

"complete quantity"

What is "complete Quantity"?

 

For example:

 

Please show us stap by stap how can we define the complete number of elements in {1,1,1,1,1,1,1,...}?

 

Thank you.

[bloodhound]

size of a collection and sum of a collection?

 

well heres an example

 

[math]\{\tfrac{1}{n^2} \colon n \in \mathbb{N}\}[/math] ovbiously there is a bijection with [math]\mathbb{N}[/math]

 

and the sum of the elements of that set is equal to [math]\frac{\pi^2}{6}[/math]

[JaKiri]

It does not give any information about the number of these elements, thesefore you cannot use the word 'ALL', because it gives you the illusion that you can know the Size of this collection, and I do not see how you can define this Size.

 

Oh sweet jesus. That's not only wrong on an actual level but a fundamental one as well.

[lama]

size of a collection and sum of a collection?

 

well heres an example

 

[math]\{\tfrac{1}{n^2} \colon n \in \mathbb{N}\}[/math] ovbiously there is a bijection with [math]\mathbb{N}[/math]

 

and the sum of the elements of that set is equal to [math]\frac{\pi^2}{6}[/math]

No Bloodhound, it is not ovbious at all, please read post #1 to see it for yourself.

[VazScep]

One of the most devastating things in the Language of Mathematics and its logical reasoning is a hidden assumption' date=' and the worst thing is a first-order hidden assumption.

 

Let FOHA be a First-Order Hidden Assumption.

 

Let UQ be Universal-Quantification.

 

Is there a FOHA in the way we use UQ concept?

 

The UQ is based on the term ‘For All’.[/quote']It is a symbol in an idealised language whose syntax is based on the use of certain phrases and pronouns, providing a more powerful system of cross-reference for their antecedents.

 

The meaning of the word ‘All’ is synonym to the word ‘Complete’

Synonyms are interchangable in a sentence. "For all..." cannot be replaced by "For complete..."

 

and if it is related to a collection of elements, then from a quantitative point of view ‘All’ is actually the SUM of this collection, where in the level of SUM we are no longer in the level of each single element that exists in this collection.

 

The SUM of a collection is actually its Cardinality.

Are you suggesting that a sentence such as "All humans are mortal..." is actually to be understood as "The 6 billion (or so) humans are mortal"? And likewise with other such statements?

 

If we define some property that can be found in each single element in this collection, then this property cannot be SUMMERIZED, because this definition can exist only in the level of each single element.

 

For example: All men are mortal.

 

The property of being mortal does not changed by the quantity of men.

Well the class of men is not mortal, so yes, the property is not changed by the quantity of men.

 

In this case we cannot use the Quantity concept because the existence of this definition cannot be developed beyond the Quantity 1 (each single element).

Which definition?

 

So we see that there can be found some common property to several distinguished elements, but this common property is not related to the Quantity concept, and therefore it cannot be SUMMERIZED, or in other words, this common property cannot be summarized beyond the quantity 1, and the meaning of quantity 1 in this case is synonym to the term ‘There exists’.

I don't understand. "There exists a man who is mortal" is not what we want, so what does our statement have to do with "there exists..."?

 

I'll stop here. The rest confuses me just as much.

 

By the way, if the logical idioms of Hebrew drastically differ from those of English then we may have serious communication problems here.

[zaphod]

lama lama lama lama... when will you ever stop with the acid and take a decent math class?

[lama]

By the way' date=' if the logical idioms of Hebrew drastically differ from those of English then we may have serious communication problems here.

[/quote']

Dear Vazscap,

 

Is there any difference between "For each" and "For all" in English?

[VazScep]

The above property is true for each single element > 9.

 

It does not give any information about the number of these elements' date=' thesefore you cannot use the word 'ALL', because it gives you the illusion that you can know the Size of this collection, and I do not see how you can define this Size.

 

So please show us how can we define the Size of N, for example?

 

If you cannot do that, instead "For All", you have to use:

 

"There exists some arbitrary n in N such that ...."[/quote']That is just bad English. Do you mean "Let n be an abitrary member of N"? The meaning of this sentence is to specify an object satisfying no conditions other than membership in N. Thus, what follows is taken to be equally valid for any given member of N. If such statements are rendered in formal logic, they become universal quantifications.

 

Or we can invent a new Quantifeir, which its name is: "For Each" and it is equivalent to ""There exists some arbitrary ...".

"For each..." does not convey the same meaning as "There exists some abitrary...". Existence statements only guarantee a single object satisfying a condition. I have no idea what it means for an arbitrary object to exist. Is the number 3 arbitrary, for instance? Arbitrary in what sense?

 

By the way, every time you read "For all..." you should be able to replace it with "For each...". If you only have a complaint with the former, why don't you try doing this?

[bloodhound]

No Bloodhound, it is not ovbious at all, please read post #1 to see it for yourself.

 

well, if you ever bothered to use the Universally acknowledged DEFINITION of a bijection then it should be obvious to you.

[VazScep]

Dear Vazscap' date='

 

Is ther any difference between "For each" and "For all" in English?[/quote']Semantically no. Grammatically yes, and because of this, mathematicians often commit the following mistake:

 

"For all n..."

 

treating the "n" as singular, when "for all" requires a plural. If you are a grammar nazi, you might insist on saying "for each" for this reason.

[lama]

Are you suggesting that a sentence such as "All humans are mortal..." is actually to be understood as "The 6 billion (or so) humans are mortal"? And likewise with other such statements?

As I said the property of mortality is not chaneged by the number of members that can be found in some collection (finite or infinite).

 

But the Cardinality of non-finite cannot be found, because this is exactly the difference between a finite collection (its Cardinality can be found) and a non finite collection (Its Cardinality cannot be found).

 

In order to see my point of view, please read:

 

http://www.geocities.com/complementarytheory/EProp.pdf

 

Thank you.

[VazScep]

Actually, forget my attempt at a grammar lecture. I have a feeling both versions would be criticised by an English professor. Since these idioms are exclusive to mathematics and logic (we do not say "for all people, it is free", nor "for each person, it is free), I would say that both phrases are intended as a shorthand for "whatever person you choose" or "given a person" etc...

[matt grime]

In the case of 1-1 between two collections of Natural numbers' date=' then the cardinality the total number (the SUM) of 1-1 mapping that can be found.

 

Yeh, I know The SIZE of a collection is actually its Cardinality.

 

Can you please explain what is the difference between Size of a collection and Sum of a collection?

 

Thank you.[/quote']

 

 

No, doron, cardinality, whilst being in a loose sense an indicator of size, isn't actually defined as "it's size" since that isn't a mathematical statement or a definition in the mathematical sense. I will side step delicate set theoretic results you are not sufficiently well versed in (and neither am I, so don't be offended), a cardinal number isn't a number, it is an isomorphism class of objects in the category SET. When we talk about cardinality we are in effect making a statement about the isomorphism class to which the set belongs. Nothing more.

 

 

For finite sets, we can think of (natural) numbers as cardinals. THis doesn't mean anything for infinite sets, where we generalize the properties.

 

 

There is some subtle difference, as Vazcep notes, between the uses of for each and for all. However, the sentences P is true for all members of S and P is true for each member of S roughly are the same. There are also subltle differences such as:

 

let S be a set of subset of R.

 

1. there is a lower bound for each set in S

 

and

 

2. there is a lower bound for all sets in S

 

have slightly different meanings because one is singular the other plural, yet the there is is singular in both cases.

 

The first may be read as saying that there is a different lower bound for each element in S, and the second might be viewed as saying that there is a single number that is a lower bound for each (i choose each carefully there) set. this is why we have developled a careful and formal system of predicate logic, with universal quantifiers to remove any ambiguity.

 

the first is properly written as

 

[math] (\forall s \in S)( \exists L \in \mathbb{R})(s \subset (L, \infty))[/math]

 

and the second is

 

[math](\exists L \in \mathbb{R})(\forall s \in S)(s \subset (L,\infty)[/math]

[lama]

well' date=' if you ever bothered to use the Universally acknowledged DEFINITION of a bijection then it should be obvious to you.

[/quote']

I know what is 1-1 and onto, but how you can use it to define the size(s) of collections of infinitely many elements?

 

and please do not give me Cantor's second diagonal, because first show us how you find the exact value of |N|.

[lama]

I would say that both phrases are intended as a shorthand for "whatever person you choose" or "given a person" etc...

So, as you see you also related to to level of a single member and not to all members at once.

 

All members at once is meaninful only if their exact SUM also can be found, but in a non-finite collection, we can talk only in the level of a single given member.

 

A Bijection does not change this state, so how can we find the exact Cardinality of N, using a 1-1 (look for youself a bijection is based on 1-1, or in other words, on the level of a single object) and onto?

 

Furthermore, what is the meaning of onto and how can we translate it to an exact Cardinal?

[matt grime]

So' date=' as you see you also related to to level of a single member and not to all members at once.

 

All members at once is meaninful only if their exact SUM also can be found, but in a non-finite collection, we can talk only in the level of a single given member.[/quote']

 

doron, until such time as you define SUM and size what you are writing is meaningless.

[lama]

SUM is the number of elemets that can be found in a finite collection, and the word 'ALL' can be related only to finite collections.

the sentences P is true for all members of S and P is true for each member of S roughly are the same.

I agree with you but only S is a finite collection.

 

Again:

 

Please show us stap by stap how can we define the complete number of elements in {1,1,1,1,1,1,1,...}?

 

Thank you.

[VazScep]

So, as you see you also related to to level of a single member and not to all members at once.

But to say that, whatever natural number you choose, that number will be non-negative, is to say something about all natural numbers. But it does not say anything explicit about the set N.

 

All members at once is meaninful only if their exact SUM also can be found, but in a non-finite collection, we can talk only in the level of a single given member.

 

A Bijection does not change this state, so how can we find the exact Cardinality of N, using a 1-1 (look for youself a bijection is based on 1-1, or in other words, on the level of a single object) and onto?

I'm not sure I understand what you are saying, but here goes:

 

"The set N" does not mean the same thing as "all N". Statements involving "all N" are statements about the individual natural numbers, to be translated as I suggested as "whatever n in N you may choose". However, statements involving "the set N" do not have to say anything about natural numbers. For instance, I can tell you that N belongs to the set {N}, without having to discuss any given natural number.

 

Cardinality is a property of a set defined in terms of mappings, which themselves are defined in terms of individual members of the set.

[bloodhound]

I know what is 1-1 and onto' date=' but how you can use it to define the size(s) of collections of infinitely many elements?

 

[/quote']

why are u asking ME? i mean , i was just replying to ur previous statement. i had no intention of defining size of collections of infinitely many elements using a mapping.